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Unformatted text preview: ained from Fc(w) by replacing w w ith n iT. Therefore, F (n)
is identical t o Fe(w) frequency-scaled by a factor T , as shown in Fig. 1O.8f.
I n t he above derivation, we d id n ot have t o use t he a ssumption t hat fc(t) is
b andlimited o r t hat fc(t) is s ampled a t a r ate a t l east equal t o i ts Nyquist rate. I f
t he signal were not bandlimited, t he only difference in o ur discussion would be in t he
sketches in Fig. 10.8. For instance, Fe(w) in Fig. 1O.8b would n ot b e bandlimited.
This fact would cause overlapping (aliasing) of repeating cycles of Fc(w) in Figs.
lO.Sd a nd f. A similar t hing h appens when t he signal fc(t) is s ampled below t he
Nyquist r ate.t 10.4-1 D TFT o f Decimated and Interpolated Signals We can use Eq. (10.55) t o find t he D TFT o f d ecimated a nd i nterpolated sigtIn case t he signal is n ot bandlimited a nd/ot t he s ampling r ate is below its Nyquist rate, t he
samples f c(kT) c an b e interpreted as t he Nyquist samples o f t he inverse Fourier transform o f t he
first cycle (centered a t w = 0) o f T F c( w ) . 638 10 Fourier Analysis of Discrete-Time Signals nals, which are explained in Fig. 8.17. Consider a signal j [k] a nd i ts D TFT F (n),
a s illustrated in Figs. 1O.9a a nd b. Figure 1O.9c shows t he d ecimated signal f[2k].
I f f [k] is considered t o b e t he sample sequence o f a c ontinuous-time signal fc(t),
t hen f[2k] is the sample sequence of fc(2t), whose Fourier transform is given by
~Fc(w/2) according t o Eq. (4.34).t As seen in Eq. (10.55), F (n) is F c(n/T) repeating periodically with period 21r, a nd t he D TFT of f[2k] is ~Fc(n/2T) r epeating
periodically with period 21r. N ote t hat F (n/2T) is F (n/T) t ime-expanded by factor 2, as shown in Figs. 1O.9b a nd d. I f we use m th-order decimation; t hat is, if
we select every m t h element in t he sequence, t he r esulting decimated sequence will
be f [mk]. Using the above argument, it follows t hat in t he f undamental frequency
range, t he D TFT of f [mk] will be l /m t imes F (n/m), a nd i t r epeats periodically
with period 21r. I f m is t oo large, so t hat t he first cycle of F (n/m) goes beyond 1r,
t he successive cycles o f F (n/m) will overlap, as illustrated in Fig. 1O.9d.
Now consider t he i nterpolated signal J;[k] in Fig. 1O.9c. T his signal is f [k/2]
( obtained by expanding f [k] by factor 2), with t he a lternate (missing) points filled
by interpolated values obtained by ideal lowpass filtering. I f t he envelope o f f [k]
is fc(t), t hen t he envelope o f f dk] is f c(t/2), whose Fourier transform is 2Fc(2w).
T hus, in t he f undamental frequency range, t he s pectrum of J;[k] is 2 t imes F (n)
compressed by factor 2 along t he frequency axis, a nd periodically repeating with
period 21r as depicted in Fig. 10.9f. Using a similar argument, we c an generalize
this result for a time-expanded signal f [k/m] w ith missing values filled by ideal
interpolation. In this case, t he s pectrum in t he f undamental frequency ra...
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